Answer :
To solve this problem, let's reason step-by-step through the scenario of the cell phone's depreciation.
1. Initial Value: The initial value of the cell phone is [tex]$750$[/tex].
2. Depreciation Rate: The cell phone loses [tex]$28\%$[/tex] of its value each year. This means each year it retains [tex]$100\% - 28\% = 72\%$[/tex] of its value.
3. Annual Retention Percentage: If the cell phone retains [tex]$72\%$[/tex] of its value each year, the retention factor is [tex]$0.72$[/tex].
4. General Form of an Exponential Decay Function: The general form of an exponential decay function is:
[tex]\[ f(x) = A \cdot (r)^x \][/tex]
where:
- [tex]\(A\)[/tex] is the initial amount (initial value of the cell phone, which is [tex]$750$[/tex]),
- [tex]\(r\)[/tex] is the decay factor (annual retention factor, which is [tex]$0.72$[/tex]), and
- [tex]\(x\)[/tex] is the number of years.
5. Substituting the Values: Substitute [tex]$A = 750$[/tex] and [tex]$r = 0.72$[/tex] into the general form of the exponential decay function:
[tex]\[ f(x) = 750 \cdot (0.72)^x \][/tex]
Therefore, the correct exponential decay function to represent the cell phone losing [tex]$28\%$[/tex] of its value each year is:
[tex]\[ f(x) = 750 \cdot 0.72^x \][/tex]
Among the given options, the correct one is:
[tex]\[ \boxed{f(x) = 750 \cdot 0.72^x} \][/tex]
1. Initial Value: The initial value of the cell phone is [tex]$750$[/tex].
2. Depreciation Rate: The cell phone loses [tex]$28\%$[/tex] of its value each year. This means each year it retains [tex]$100\% - 28\% = 72\%$[/tex] of its value.
3. Annual Retention Percentage: If the cell phone retains [tex]$72\%$[/tex] of its value each year, the retention factor is [tex]$0.72$[/tex].
4. General Form of an Exponential Decay Function: The general form of an exponential decay function is:
[tex]\[ f(x) = A \cdot (r)^x \][/tex]
where:
- [tex]\(A\)[/tex] is the initial amount (initial value of the cell phone, which is [tex]$750$[/tex]),
- [tex]\(r\)[/tex] is the decay factor (annual retention factor, which is [tex]$0.72$[/tex]), and
- [tex]\(x\)[/tex] is the number of years.
5. Substituting the Values: Substitute [tex]$A = 750$[/tex] and [tex]$r = 0.72$[/tex] into the general form of the exponential decay function:
[tex]\[ f(x) = 750 \cdot (0.72)^x \][/tex]
Therefore, the correct exponential decay function to represent the cell phone losing [tex]$28\%$[/tex] of its value each year is:
[tex]\[ f(x) = 750 \cdot 0.72^x \][/tex]
Among the given options, the correct one is:
[tex]\[ \boxed{f(x) = 750 \cdot 0.72^x} \][/tex]
